| L(s) = 1 | + (0.996 + 0.0871i)2-s + (0.984 + 0.173i)4-s + (1.16 + 1.90i)5-s + (1.36 + 0.956i)7-s + (0.965 + 0.258i)8-s + (0.992 + 2.00i)10-s + (−1.17 + 3.23i)11-s + (−0.0285 − 0.326i)13-s + (1.27 + 1.07i)14-s + (0.939 + 0.342i)16-s + (−2.17 + 0.583i)17-s + (−0.248 + 0.143i)19-s + (0.814 + 2.08i)20-s + (−1.45 + 3.12i)22-s + (−0.780 − 1.11i)23-s + ⋯ |
| L(s) = 1 | + (0.704 + 0.0616i)2-s + (0.492 + 0.0868i)4-s + (0.520 + 0.853i)5-s + (0.516 + 0.361i)7-s + (0.341 + 0.0915i)8-s + (0.313 + 0.633i)10-s + (−0.355 + 0.975i)11-s + (−0.00793 − 0.0906i)13-s + (0.341 + 0.286i)14-s + (0.234 + 0.0855i)16-s + (−0.528 + 0.141i)17-s + (−0.0569 + 0.0328i)19-s + (0.182 + 0.465i)20-s + (−0.310 + 0.665i)22-s + (−0.162 − 0.232i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.24731 + 1.33981i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.24731 + 1.33981i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.996 - 0.0871i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.16 - 1.90i)T \) |
| good | 7 | \( 1 + (-1.36 - 0.956i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (1.17 - 3.23i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.0285 + 0.326i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (2.17 - 0.583i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.248 - 0.143i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.780 + 1.11i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-4.61 + 3.86i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.40 + 7.98i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.38 - 5.16i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (5.63 - 6.71i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.15 - 4.61i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-4.47 + 6.39i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-8.21 + 8.21i)T - 53iT^{2} \) |
| 59 | \( 1 + (-10.1 + 3.67i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.55 - 8.81i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (13.6 - 1.19i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-4.96 - 2.86i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.683 - 2.55i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (1.13 + 1.35i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.19 + 13.6i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (0.603 + 1.04i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-15.8 + 7.39i)T + (62.3 - 74.3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.26428418112036934971883342266, −9.932309216063582243004730013822, −8.574367919627898005895366772512, −7.65404371907577598996609784240, −6.77205133752413128248288069057, −6.01824958627161436838578081507, −5.03583275596887230645695941406, −4.12699335907499106960437895922, −2.74045653847210845260283693540, −1.98857821841865972765385256013,
1.11732754056158494028308073272, 2.46868361816543887379766340904, 3.79361227900779912600907920172, 4.82144013467451230719564385289, 5.47562917260583039819848091486, 6.41622193583756671051326300135, 7.47541320990176402064326792849, 8.511074925183724955873691170903, 9.098637947758200001796603450642, 10.41314771524992503642572851927
